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Chapter 4: Problem 28

Write the equation in logarithmic form. $$ 2^{5}=32 $$

Short Answer

Expert verified
log_2(32) = 5

Step by step solution

01

Identify the Elements

Recognize the base, exponent, and result from the given exponential equation. Here, the base is 2, the exponent is 5, and the result is 32.
02

Apply the Logarithmic Form

Remember that the logarithmic form of an exponential equation such as 2^5=32 is given by: log_base(result) = exponent. Therefore, we substitute base=2, result=32, and exponent=5.
03

Write the Logarithmic Equation

Using the formula from Step 2, we write log_2(32) = 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithms
Logarithms might seem confusing at first, but they're just another way to express exponential equations. In essence, a logarithm asks the question: 'What power do we need to raise a given base to, in order to get a certain number?' For example, if we have the equation \(2^{5}=32\), we can rewrite it as a logarithmic equation: \(\text{log}_{2}(32)=5\). This means that if you raise the base 2 to the power of 5, you get 32.
Exponential Equations
Exponential equations are equations where the variable is in the exponent. For instance, in the equation \(2^{5}=32\), 2 is the base, 5 is the exponent, and 32 is the result. Exponential equations can always be converted into logarithmic form, which can often make solving them easier. To convert, remember the relationship: \(b^{e}=r\) converts to \(\text{log}_{b}(r)=e\). Understanding this conversion is key to working with both forms.
Base and Exponent
When working with exponential and logarithmic forms, it's crucial to identify the base and the exponent correctly. In the given problem \(2^{5}=32\):
  • The base is 2. This is the number that's being raised to a power.
  • The exponent is 5. This indicates how many times the base is multiplied by itself.
  • The result is 32. This is what you get after performing the exponentiation.
From this, the corresponding logarithmic form is \(\text{log}_{2}(32)=5\), showing that 2 raised to the power of 5 equals 32.

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Most popular questions from this chapter

Prove the power property of logarithms: \(\log _{b} x^{p}=p \log _{b} x\)

\(F(C)=\frac{9}{5} C+32\) gives the temperature in degrees Fahrenheit as a function of the temperature \(C\) in degrees Celsius. Find an equation for \(C(F)\) and interpret its meaning in the context of this problem.

Al needs to borrow \(\$ 15,000\) to buy a car. He can borrow the money at \(6.7 \%\) simple interest for 5 yr or he can borrow at \(6.4 \%\) interest compounded continuously for 5 yr. a. How much total interest would Al pay at \(6.7 \%\) simple interest? b. How much total interest would Al pay at \(6.4 \%\) interest compounded continuously? c. Which option results in less total interest?

In \(1989,\) the Loma Prieta earthquake damaged the city of San Francisco with an intensity of approximately \(10^{6.9} I_{0} .\) Film footage of the 1989 earthquake was captured on a number of video cameras including a broadcast of Game 3 of the World Series played at Candlestick Park. (See Example \(\mathbf{1 0}\) ) a. Determine the magnitude of the Loma Prieta earthquake. b. Smaller earthquakes occur daily in the San Francisco area and most are not detectable without a seismograph. Determine the magnitude of an earthquake with an intensity of \(10^{3.2} I_{0}\). c. How many times more intense was the Loma Prieta earthquake than an earthquake with a magnitude of 3.2 on the Richter scale? Round to the nearest whole unit.

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(80=320 e^{-0.5 t}\)
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